The following references, to the extent that they provide exemplary procedural or other details supplementary to those set forth herein, are specifically incorporated herein by reference. These references may provide certain background regarding the subject matter discussed herein.
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It has been a desire of scientists to develop control mechanisms so that machines could function independently without human intervention. Such controlled machines should be able to complete an unstructured task and learn from the feedback information about their performance. These machines, therefore, should be able to learn tasks not easily handled by existing machines, and more importantly, continue to adapt and perform these tasks with increasing efficacy under uncertainties.
In order to confront modern technological problems that require systems with intelligent functions such as simultaneous utilization of memory, learning, or high-level decision making in response to "fuzzy" or qualitative commands, intelligent controls is being investigated. Intelligent control should utilize cognitive theory effectively with various mathematical programming techniques. Learning is a first step toward intelligent control and would replace the human operator by making intelligent choices whenever the environment does not allow or justify the presence of a human operator. Learning has the capability of reducing the uncertainties affecting the performance of a dynamical system through on-line modeling (system identification), thereby improving the knowledge about the system so that it can be controlled more effectively.
Considerable research has been conducted in system identification (Narendra and Parthasarathy, 1990) or identification-based NN control (Levine and Narendra, 1993), and little about the use of direct closed-loop multilayer NN controllers that yield guaranteed performance (Chen and Khalil, 1992). On the other hand, some results presenting the relations between NN and direct adaptive control (Landau, 1993; Lewis et al., 1993), as well as some notions on NN for robot control, are given in (Sadegh, 1991; Sanner and Slotine, 1991). A direct continuous-time multilayer neural net robot controller was proposed in (Levine and Narendra, 1993) which guarantees closed-loop tracking performance. However, little about the application of discrete-time multilayer NN in direct closed-loop controllers that yield guaranteed performance is discussed in the literature.
The controller design with NN having multilayers for both continuous an d discrete-time is treated in (Chen and Khalil, 1992; Lewis et al., 1993; Sira-Ramirez and Zak, 1991). In (Chen and Khalil, 1992), the adaptive control of nonlinear systems using multilayer NN is presented very nicely. However, the performance of this controller is dependent upon the choice of the deadzone and the richness of the input signal. In addition, an explicit learning phase for the NN controller is needed initially. In (Sira-Ramirez and Zak, 1991), it is assumed that the input to the multilayer NN is considered to be fixed for successive iterations which is an unreasonable assumption for the controller design. Furthermore, in these papers (Chen and Khalil, 1992; Sira-Ramirez and Zak, 1991) passivity properties of the NN are not investigated. A three-layer NN controller design is presented in (Lewis et al., 1993) for the control of continuous-time systems. However, generalization of the stability analysis to NN having arbitrary number of hidden layers cannot be deduced due to the problem of defining and verifying the persistency of excitation condition for a multilayer NN. In addition, the weight tuning algorithms presented and the associated stability analysis discussed in (Lewis et al., 1993) is specific to robotic systems.
To confront all these issues head on in this invention, a family of novel learning schemes is investigated for a multilayer discrete-time NN whose weights are tuned on-line with no learning phase needed. The weight tuning mechanisms guarantee convergence of the NN weights when initialized at zero, even though there do not exist "ideal" weights such that the NN perfectly reconstructs a certain required function. The controller structure ensures good tracking performance, as shown through a Lyapunov's approach, so that the convergence to a stable solution is guaranteed. Finally, in contrast to adaptive control, it is not necessary to know a priori the structure of the plant; this structural information is instead inferred on-line by the NN.
The controller is composed of a neural net incorporated into a dynamical system, where the structure comes from filtered error notions standard in robot control literature. It is shown that the weight tuning algorithm using the standard backpropagation delta rule in each layer a passive neural net. This, if coupled with the dissipativity of the dynamical system, guarantees the boundedness of all the signals in the closed-loop system under a persistency of excitation (PE) condition disclosed below. However, PE is difficult to guarantee in a NN, which by design has redundant parameters for robust performance. Unfortunately, if PE does not hold, the delta rule generally does not guarantee tracking and bounded weights. Moreover, it is found here that the maximum permissible tuning rate for the developed tuning algorithm decreases as the NN size increases; this is a major drawback. A projection algorithm discussed herein is shown to easily correct the problem. Finally, new modified weight tuning algorithms introduced avoid the need for PE by making the NN robust, that is, state strict passive.